Besanko and Braeutigam, CH 11
Western University
When an individual agent can affect the price (or other outcome) that prevails in the market, the agent has market power
A monopoly market consists of a single seller facing many buyers
A monopsony market consists of a single buyer facing many sellers
In contrast to a perfectly competitive firm, a monopolist sets the market price of its product (market power)
The demand curve stops the monopolist from setting an infinitely high price by imposing a trade-off
The monopolist’s demand curve is the market demand curve
The profit-maximizing monopolist’s problem is finding the optimal trade-off between volume and margin (difference between price and marginal cost)
The monopolist’s profit-maximization problem : \[ \begin{aligned} \max_Q \pi(Q)&=TR(Q)-TC(Q) \\ \text{s.t. }& TR(Q)=QP(Q) \end{aligned} \qquad(1)\]
Profits are maximized at \(Q^*\) such that \(MR(Q^*)=MC(Q^*)\)
Slope of the \(MC\) curve exceeds the slope of the \(MR\) curve
Competitive firms is not affected by the change in price due to its change in output
Marginal revenue has two parts:
The marginal revenue is less than the price the monopolist can charge to sell that quantity for any \(Q\ge0\) because \(dP(Q)/dQ\le0\)
MR can be positive or negative
Suppose the monopolist faces a linear demand curve
\[ P(Q)=a-bQ \]
Then, the revenue function is
\[ TR(Q)=QP(Q)=aQ-bQ^2 \]
and the marginal revenue function is
\[ MR(Q)=a-2bQ \]
Pareto Efficiency: There is no way to make someone better off without making somebody else worse off
Inverse demand curve: At each level of output, \(P(Q)\) measures how much people are willing to pay for an additional unit of the good
Since \(P(Q)\ge MC(Q)\) for all output levels between \(Q_M\) and \(Q_C\)
There is a range of consumers that are willing to pay \(\bar P\) for an extra unit of output, such that \(P(Q)>\bar P> MC(Q)\) (area under demand curve between \(Q_M\) and \(Q_C\) and above the \(MC(Q)\))
Any of these consumers would be better off because they are willing to pay \(P(Q)\) but the extra unit of output is sold at \(\bar P < P(Q)\)
Likewise, the firm would be better off because it cost \(MC(Q)\) to produce the extra unit of output and the firms sold it for \(\bar P > MC(Q)\)(all the other units of output are being sold for the same price \(P_M\) as before)
We found a Pareto improvement!
How inefficient is a Monopoly?
Compare changes in producer’s and consumer’s surplus from a movement from Monopoly to Perfect Competition
Change in producer’s surplus —firm’s profits— measures how much the owners would be willing to pay to get the higher price under monopoly
Change in consumers’ surplus measures how much the consumers would have to be paid to compensate them for the higher price
Producer loses A, but gains C
Consumer gains A and B
A is just a transfer; total surplus does not change
B+C is a real gain in surplus
Deadweight Loss of Monopoly (B+C) measures how much worse off people are paying the monopoly price than paying the competitive price
DWL is equal to the value of loss output by valuing each unit of lost output at the price that people are willing to pay for that unit
In Competitive Markets, \(P=MC(Q)\), there is a unique relationship between the quantity produced by the firm and price \(Q=MC^{-1}(P)\)
In Monopoly, \(P+Q\frac{dP(Q)}{dQ}=MC(Q)\)
Depending on the shape of the demand curve,\(\frac{dP(Q)}{dQ}\), the monopolist might produce the same quantity at two different prices or produce different quantities at the same price
\[ \begin{aligned} MR(Q)&=P+Q\frac{dP(Q)}{dQ} \\ &=P\left(1+\frac{Q}{P}\frac{dP(Q)}{dQ} \right)\\ &=P\left(1+\frac{1}{\frac{P}{Q}\frac{dQ}{dP}}\right)\\ &=P\left(1+\frac{1}{\epsilon_{Q,P}}\right) \\ &=P\left(1-\frac{1}{|\epsilon_{Q,P}|}\right) \end{aligned} \]
Last line follows because price elasticity of demand \(\epsilon_{Q,P}<0\)
| Elasticity | Marginal Revenue | Output and Profits |
|---|---|---|
| \(|\epsilon_{Q,P}|<1\) | \(MR<0\) | Reducing output increases revenue and reduces cost, so profits necessarily increase |
| \(|\epsilon_{Q,P}|\ge1\) | \(MR\ge 0\) | Increasing output increases revenue but cost increase, optimal output lies here |
| \(|\epsilon_{Q,P}|=\infty\) | \(MR=P\) | Competitive case |
\(MC\ge0\), at the optimum \(MR=MC\ge0\), but \(|\epsilon_{Q,P}|<1 \implies MR <0\)
Any point were \(|\epsilon_{Q,P}|<1\) cannot be a profit maximum, since the monopolist could increase profits by producing less output
Profit maximizing condition is \(MR = MC\) with \(P^*\) and \(Q^*\)
\[ \begin{aligned} 𝑀𝑅(𝑄^∗ )&=𝑀𝐶(𝑄^∗ ) \\ 𝑃^∗ \left(1-\frac{1}{|\epsilon_{𝑄,𝑃}|}\right)&=𝑀𝐶(𝑄^∗) \end{aligned} \]
Rearranging and setting MR(Q) = MC(Q) \[ \frac{(𝑃^∗−𝑀C^∗)}{𝑃^∗} =\frac{1}{ |\epsilon_{𝑄,𝑃}| } \]
Inverse elasticity pricing rule (IEPR): The less price elastic the demand, the higher the optimal markup
Market B is less price elastic than A, thus the markup is higher in B than in A
When a firm can exercise some degree of control over its price in the market, we say that it has market power
Monopolists or producers of differentiated products will, in general, charge prices that exceed their marginal cost
A natural measure of market power is \((P − MC)/P\)
The Lerner Index is zero for a perfectly competitive industry. It is positive for any industry that departs from perfect competition.
The IEPR tells us that in the equilibrium of a monopoly market, the Lerner Index will be inversely related to the market price elasticity of demand.
An important driver of the price elasticity of demand is the threat of substitute products outside the industry
If a monopoly market faces strong competition from substitute products, the Lerner Index can still be low. In other words, a firm might have a monopoly, but its market power might still be weak.
Read Ch. 11.6
Natural Monopolies
Barriers to entry
A monopsony market consists of a single buyer facing many sellers
The monopsonist’s profit maximization problem:
The monopsonist’s profit maximization condition:
Consumer (Monopsony firm) gets A+B+C
Producer (Workers) gets D
The deadweight loss is F+G
Shifts in demand
Shifts in MC
Constant MC and linear demand (Monopoly Midpoint Pricing Rule)